Quadratic roots, narrow range

Percentage Accurate: 55.4% → 91.2%

Time: 30.3s

Alternatives: 9

Speedup: 29.0×

Specification

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 55.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 91.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(-4 \cdot a + \frac{{b}^{2}}{c}\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\frac{c \cdot c}{{\left(-b\right)}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.0)
   (/ (- (sqrt (* c (+ (* -4.0 a) (/ (pow b 2.0) c)))) b) (* 2.0 a))
   (-
    (*
     a
     (+
      (/ (* c c) (pow (- b) 3.0))
      (*
       a
       (+
        (* -2.0 (/ (pow c 3.0) (pow b 5.0)))
        (* -5.0 (/ (* a (pow c 4.0)) (pow b 7.0)))))))
    (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.0) {
		tmp = (sqrt((c * ((-4.0 * a) + (pow(b, 2.0) / c)))) - b) / (2.0 * a);
	} else {
		tmp = (a * (((c * c) / pow(-b, 3.0)) + (a * ((-2.0 * (pow(c, 3.0) / pow(b, 5.0))) + (-5.0 * ((a * pow(c, 4.0)) / pow(b, 7.0))))))) - (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 2.0d0) then
        tmp = (sqrt((c * (((-4.0d0) * a) + ((b ** 2.0d0) / c)))) - b) / (2.0d0 * a)
    else
        tmp = (a * (((c * c) / (-b ** 3.0d0)) + (a * (((-2.0d0) * ((c ** 3.0d0) / (b ** 5.0d0))) + ((-5.0d0) * ((a * (c ** 4.0d0)) / (b ** 7.0d0))))))) - (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.0) {
		tmp = (Math.sqrt((c * ((-4.0 * a) + (Math.pow(b, 2.0) / c)))) - b) / (2.0 * a);
	} else {
		tmp = (a * (((c * c) / Math.pow(-b, 3.0)) + (a * ((-2.0 * (Math.pow(c, 3.0) / Math.pow(b, 5.0))) + (-5.0 * ((a * Math.pow(c, 4.0)) / Math.pow(b, 7.0))))))) - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.0:
		tmp = (math.sqrt((c * ((-4.0 * a) + (math.pow(b, 2.0) / c)))) - b) / (2.0 * a)
	else:
		tmp = (a * (((c * c) / math.pow(-b, 3.0)) + (a * ((-2.0 * (math.pow(c, 3.0) / math.pow(b, 5.0))) + (-5.0 * ((a * math.pow(c, 4.0)) / math.pow(b, 7.0))))))) - (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.0)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(Float64(-4.0 * a) + Float64((b ^ 2.0) / c)))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(a * Float64(Float64(Float64(c * c) / (Float64(-b) ^ 3.0)) + Float64(a * Float64(Float64(-2.0 * Float64((c ^ 3.0) / (b ^ 5.0))) + Float64(-5.0 * Float64(Float64(a * (c ^ 4.0)) / (b ^ 7.0))))))) - Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.0)
		tmp = (sqrt((c * ((-4.0 * a) + ((b ^ 2.0) / c)))) - b) / (2.0 * a);
	else
		tmp = (a * (((c * c) / (-b ^ 3.0)) + (a * ((-2.0 * ((c ^ 3.0) / (b ^ 5.0))) + (-5.0 * ((a * (c ^ 4.0)) / (b ^ 7.0))))))) - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.0], N[(N[(N[Sqrt[N[(c * N[(N[(-4.0 * a), $MachinePrecision] + N[(N[Power[b, 2.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(N[(c * c), $MachinePrecision] / N[Power[(-b), 3.0], $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-5.0 * N[(N[(a * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2

   1. Initial program 86.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 86.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

      2. +-commutative 86.0%

         \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]

      3. sqr-neg 86.0%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{a \cdot 2} \]

      4. unsub-neg 86.0%

         \[\leadsto \frac{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right) - \left(4 \cdot a\right) \cdot c} - b}}{a \cdot 2} \]

      5. sqr-neg 86.0%

         \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b} - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \]

      6. fma-neg 86.0%

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}} - b}{a \cdot 2} \]

      7. distribute-lft-neg-in 86.0%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-4 \cdot a\right) \cdot c}\right)} - b}{a \cdot 2} \]

      8. *-commutative 86.0%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-4 \cdot a\right)}\right)} - b}{a \cdot 2} \]

      9. *-commutative 86.0%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-\color{blue}{a \cdot 4}\right)\right)} - b}{a \cdot 2} \]

      10. distribute-rgt-neg-in 86.0%

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(a \cdot \left(-4\right)\right)}\right)} - b}{a \cdot 2} \]

      11. metadata-eval 86.0%

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot \color{blue}{-4}\right)\right)} - b}{a \cdot 2} \]

   3. Simplified 86.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in c around inf 86.0%

      \[\leadsto \frac{\sqrt{\color{blue}{c \cdot \left(-4 \cdot a + \frac{{b}^{2}}{c}\right)}} - b}{a \cdot 2} \]

   if 2 < b

   1. Initial program 45.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 45.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

      2. +-commutative 45.8%

         \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]

      3. sqr-neg 45.8%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{a \cdot 2} \]

      4. unsub-neg 45.8%

         \[\leadsto \frac{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right) - \left(4 \cdot a\right) \cdot c} - b}}{a \cdot 2} \]

      5. sqr-neg 45.8%

         \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b} - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \]

      6. fma-neg 45.9%

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}} - b}{a \cdot 2} \]

      7. distribute-lft-neg-in 45.9%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-4 \cdot a\right) \cdot c}\right)} - b}{a \cdot 2} \]

      8. *-commutative 45.9%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-4 \cdot a\right)}\right)} - b}{a \cdot 2} \]

      9. *-commutative 45.9%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-\color{blue}{a \cdot 4}\right)\right)} - b}{a \cdot 2} \]

      10. distribute-rgt-neg-in 45.9%

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(a \cdot \left(-4\right)\right)}\right)} - b}{a \cdot 2} \]

      11. metadata-eval 45.9%

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot \color{blue}{-4}\right)\right)} - b}{a \cdot 2} \]

   3. Simplified 45.9%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 95.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

   6. Taylor expanded in c around 0 95.5%

      \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \color{blue}{-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}}\right)\right) \]
   7. Step-by-step derivation

      1. associate-*r/ 95.5%

         \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(\color{blue}{\frac{-1 \cdot {c}^{2}}{{b}^{3}}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right) \]

   8. Applied egg-rr 95.5%

      \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(\color{blue}{\frac{-1 \cdot {c}^{2}}{{b}^{3}}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r/ 95.5%

         \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(\color{blue}{-1 \cdot \frac{{c}^{2}}{{b}^{3}}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right) \]

      2. mul-1-neg 95.5%

         \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(\color{blue}{\left(-\frac{{c}^{2}}{{b}^{3}}\right)} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right) \]

      3. distribute-neg-frac2 95.5%

         \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(\color{blue}{\frac{{c}^{2}}{-{b}^{3}}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right) \]

      4. cube-neg 95.5%

         \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(\frac{{c}^{2}}{\color{blue}{{\left(-b\right)}^{3}}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right) \]

   10. Simplified 95.5%

       \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(\color{blue}{\frac{{c}^{2}}{{\left(-b\right)}^{3}}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right) \]
   11. Step-by-step derivation

       1. unpow2 95.5%

          \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(\frac{\color{blue}{c \cdot c}}{{\left(-b\right)}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right) \]

   12. Applied egg-rr 95.5%

       \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(\frac{\color{blue}{c \cdot c}}{{\left(-b\right)}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(-4 \cdot a + \frac{{b}^{2}}{c}\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\frac{c \cdot c}{{\left(-b\right)}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right) - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(-4 \cdot a + \frac{{b}^{2}}{c}\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, a \cdot \frac{-2 \cdot {c}^{3}}{{b}^{4}} - {\left(\frac{c}{b}\right)}^{2}, -c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.0)
   (/ (- (sqrt (* c (+ (* -4.0 a) (/ (pow b 2.0) c)))) b) (* 2.0 a))
   (/
    (fma
     a
     (- (* a (/ (* -2.0 (pow c 3.0)) (pow b 4.0))) (pow (/ c b) 2.0))
     (- c))
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.0) {
		tmp = (sqrt((c * ((-4.0 * a) + (pow(b, 2.0) / c)))) - b) / (2.0 * a);
	} else {
		tmp = fma(a, ((a * ((-2.0 * pow(c, 3.0)) / pow(b, 4.0))) - pow((c / b), 2.0)), -c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.0)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(Float64(-4.0 * a) + Float64((b ^ 2.0) / c)))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(fma(a, Float64(Float64(a * Float64(Float64(-2.0 * (c ^ 3.0)) / (b ^ 4.0))) - (Float64(c / b) ^ 2.0)), Float64(-c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.0], N[(N[(N[Sqrt[N[(c * N[(N[(-4.0 * a), $MachinePrecision] + N[(N[Power[b, 2.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(a * N[(N[(-2.0 * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + (-c)), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2

   1. Initial program 86.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 86.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

      2. +-commutative 86.0%

         \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]

      3. sqr-neg 86.0%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{a \cdot 2} \]

      4. unsub-neg 86.0%

         \[\leadsto \frac{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right) - \left(4 \cdot a\right) \cdot c} - b}}{a \cdot 2} \]

      5. sqr-neg 86.0%

         \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b} - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \]

      6. fma-neg 86.0%

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}} - b}{a \cdot 2} \]

      7. distribute-lft-neg-in 86.0%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-4 \cdot a\right) \cdot c}\right)} - b}{a \cdot 2} \]

      8. *-commutative 86.0%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-4 \cdot a\right)}\right)} - b}{a \cdot 2} \]

      9. *-commutative 86.0%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-\color{blue}{a \cdot 4}\right)\right)} - b}{a \cdot 2} \]

      10. distribute-rgt-neg-in 86.0%

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(a \cdot \left(-4\right)\right)}\right)} - b}{a \cdot 2} \]

      11. metadata-eval 86.0%

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot \color{blue}{-4}\right)\right)} - b}{a \cdot 2} \]

   3. Simplified 86.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in c around inf 86.0%

      \[\leadsto \frac{\sqrt{\color{blue}{c \cdot \left(-4 \cdot a + \frac{{b}^{2}}{c}\right)}} - b}{a \cdot 2} \]

   if 2 < b

   1. Initial program 45.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 45.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

      2. +-commutative 45.8%

         \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]

      3. sqr-neg 45.8%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{a \cdot 2} \]

      4. unsub-neg 45.8%

         \[\leadsto \frac{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right) - \left(4 \cdot a\right) \cdot c} - b}}{a \cdot 2} \]

      5. sqr-neg 45.8%

         \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b} - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \]

      6. fma-neg 45.9%

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}} - b}{a \cdot 2} \]

      7. distribute-lft-neg-in 45.9%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-4 \cdot a\right) \cdot c}\right)} - b}{a \cdot 2} \]

      8. *-commutative 45.9%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-4 \cdot a\right)}\right)} - b}{a \cdot 2} \]

      9. *-commutative 45.9%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-\color{blue}{a \cdot 4}\right)\right)} - b}{a \cdot 2} \]

      10. distribute-rgt-neg-in 45.9%

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(a \cdot \left(-4\right)\right)}\right)} - b}{a \cdot 2} \]

      11. metadata-eval 45.9%

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot \color{blue}{-4}\right)\right)} - b}{a \cdot 2} \]

   3. Simplified 45.9%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 93.2%

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 61.5%

         \[\leadsto \frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}}{b} \]

      2. expm1-undefine 51.9%

         \[\leadsto \frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \color{blue}{\left(e^{\mathsf{log1p}\left(-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)} - 1\right)}}{b} \]

      3. neg-mul-1 51.9%

         \[\leadsto \frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(e^{\mathsf{log1p}\left(\color{blue}{\left(-c\right)} + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)} - 1\right)}{b} \]

      4. +-commutative 51.9%

         \[\leadsto \frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(e^{\mathsf{log1p}\left(\color{blue}{-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \left(-c\right)}\right)} - 1\right)}{b} \]

      5. fma-define 51.9%

         \[\leadsto \frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(-1, \frac{a \cdot {c}^{2}}{{b}^{2}}, -c\right)}\right)} - 1\right)}{b} \]

      6. associate-/l* 51.9%

         \[\leadsto \frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(-1, \color{blue}{a \cdot \frac{{c}^{2}}{{b}^{2}}}, -c\right)\right)} - 1\right)}{b} \]

   7. Applied egg-rr 51.9%

      \[\leadsto \frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \color{blue}{\left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(-1, a \cdot \frac{{c}^{2}}{{b}^{2}}, -c\right)\right)} - 1\right)}}{b} \]
   8. Step-by-step derivation

      1. sub-neg 51.9%

         \[\leadsto \frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \color{blue}{\left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(-1, a \cdot \frac{{c}^{2}}{{b}^{2}}, -c\right)\right)} + \left(-1\right)\right)}}{b} \]

      2. metadata-eval 51.9%

         \[\leadsto \frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(-1, a \cdot \frac{{c}^{2}}{{b}^{2}}, -c\right)\right)} + \color{blue}{-1}\right)}{b} \]

      3. +-commutative 51.9%

         \[\leadsto \frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \color{blue}{\left(-1 + e^{\mathsf{log1p}\left(\mathsf{fma}\left(-1, a \cdot \frac{{c}^{2}}{{b}^{2}}, -c\right)\right)}\right)}}{b} \]

      4. log1p-undefine 51.9%

         \[\leadsto \frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 + e^{\color{blue}{\log \left(1 + \mathsf{fma}\left(-1, a \cdot \frac{{c}^{2}}{{b}^{2}}, -c\right)\right)}}\right)}{b} \]

      5. rem-exp-log 83.5%

         \[\leadsto \frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 + \color{blue}{\left(1 + \mathsf{fma}\left(-1, a \cdot \frac{{c}^{2}}{{b}^{2}}, -c\right)\right)}\right)}{b} \]

      6. fma-undefine 83.5%

         \[\leadsto \frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 + \left(1 + \color{blue}{\left(-1 \cdot \left(a \cdot \frac{{c}^{2}}{{b}^{2}}\right) + \left(-c\right)\right)}\right)\right)}{b} \]

      7. neg-mul-1 83.5%

         \[\leadsto \frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 + \left(1 + \left(\color{blue}{\left(-a \cdot \frac{{c}^{2}}{{b}^{2}}\right)} + \left(-c\right)\right)\right)\right)}{b} \]

      8. distribute-neg-in 83.5%

         \[\leadsto \frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 + \left(1 + \color{blue}{\left(-\left(a \cdot \frac{{c}^{2}}{{b}^{2}} + c\right)\right)}\right)\right)}{b} \]

      9. +-commutative 83.5%

         \[\leadsto \frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 + \left(1 + \left(-\color{blue}{\left(c + a \cdot \frac{{c}^{2}}{{b}^{2}}\right)}\right)\right)\right)}{b} \]

      10. unsub-neg 83.5%

          \[\leadsto \frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 + \color{blue}{\left(1 - \left(c + a \cdot \frac{{c}^{2}}{{b}^{2}}\right)\right)}\right)}{b} \]

      11. +-commutative 83.5%

          \[\leadsto \frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 + \left(1 - \color{blue}{\left(a \cdot \frac{{c}^{2}}{{b}^{2}} + c\right)}\right)\right)}{b} \]

      12. fma-define 83.5%

          \[\leadsto \frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 + \left(1 - \color{blue}{\mathsf{fma}\left(a, \frac{{c}^{2}}{{b}^{2}}, c\right)}\right)\right)}{b} \]

   9. Simplified 83.5%

      \[\leadsto \frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \color{blue}{\left(-1 + \left(1 - \mathsf{fma}\left(a, {\left(\frac{c}{-b}\right)}^{2}, c\right)\right)\right)}}{b} \]

   10. Taylor expanded in c around 0 93.0%

       \[\leadsto \frac{\color{blue}{c \cdot \left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} - \frac{a}{{b}^{2}}\right) - 1\right)}}{b} \]

   11. Taylor expanded in a around 0 93.2%

       \[\leadsto \frac{\color{blue}{-1 \cdot c + a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{4}} + -1 \cdot \frac{{c}^{2}}{{b}^{2}}\right)}}{b} \]
   12. Step-by-step derivation

       1. neg-mul-1 93.2%

          \[\leadsto \frac{\color{blue}{\left(-c\right)} + a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{4}} + -1 \cdot \frac{{c}^{2}}{{b}^{2}}\right)}{b} \]

       2. +-commutative 93.2%

          \[\leadsto \frac{\color{blue}{a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{4}} + -1 \cdot \frac{{c}^{2}}{{b}^{2}}\right) + \left(-c\right)}}{b} \]

       3. fma-define 93.2%

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(a, -2 \cdot \frac{a \cdot {c}^{3}}{{b}^{4}} + -1 \cdot \frac{{c}^{2}}{{b}^{2}}, -c\right)}}{b} \]

       4. mul-1-neg 93.2%

          \[\leadsto \frac{\mathsf{fma}\left(a, -2 \cdot \frac{a \cdot {c}^{3}}{{b}^{4}} + \color{blue}{\left(-\frac{{c}^{2}}{{b}^{2}}\right)}, -c\right)}{b} \]

       5. unpow2 93.2%

          \[\leadsto \frac{\mathsf{fma}\left(a, -2 \cdot \frac{a \cdot {c}^{3}}{{b}^{4}} + \left(-\frac{\color{blue}{c \cdot c}}{{b}^{2}}\right), -c\right)}{b} \]

       6. unpow2 93.2%

          \[\leadsto \frac{\mathsf{fma}\left(a, -2 \cdot \frac{a \cdot {c}^{3}}{{b}^{4}} + \left(-\frac{c \cdot c}{\color{blue}{b \cdot b}}\right), -c\right)}{b} \]

       7. times-frac 93.2%

          \[\leadsto \frac{\mathsf{fma}\left(a, -2 \cdot \frac{a \cdot {c}^{3}}{{b}^{4}} + \left(-\color{blue}{\frac{c}{b} \cdot \frac{c}{b}}\right), -c\right)}{b} \]

       8. unpow2 93.2%

          \[\leadsto \frac{\mathsf{fma}\left(a, -2 \cdot \frac{a \cdot {c}^{3}}{{b}^{4}} + \left(-\color{blue}{{\left(\frac{c}{b}\right)}^{2}}\right), -c\right)}{b} \]

       9. unsub-neg 93.2%

          \[\leadsto \frac{\mathsf{fma}\left(a, \color{blue}{-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{4}} - {\left(\frac{c}{b}\right)}^{2}}, -c\right)}{b} \]

       10. *-commutative 93.2%

           \[\leadsto \frac{\mathsf{fma}\left(a, \color{blue}{\frac{a \cdot {c}^{3}}{{b}^{4}} \cdot -2} - {\left(\frac{c}{b}\right)}^{2}, -c\right)}{b} \]

       11. associate-/l* 93.2%

           \[\leadsto \frac{\mathsf{fma}\left(a, \color{blue}{\left(a \cdot \frac{{c}^{3}}{{b}^{4}}\right)} \cdot -2 - {\left(\frac{c}{b}\right)}^{2}, -c\right)}{b} \]

       12. associate-*l* 93.2%

           \[\leadsto \frac{\mathsf{fma}\left(a, \color{blue}{a \cdot \left(\frac{{c}^{3}}{{b}^{4}} \cdot -2\right)} - {\left(\frac{c}{b}\right)}^{2}, -c\right)}{b} \]

       13. *-commutative 93.2%

           \[\leadsto \frac{\mathsf{fma}\left(a, a \cdot \color{blue}{\left(-2 \cdot \frac{{c}^{3}}{{b}^{4}}\right)} - {\left(\frac{c}{b}\right)}^{2}, -c\right)}{b} \]

       14. associate-*r/ 93.2%

           \[\leadsto \frac{\mathsf{fma}\left(a, a \cdot \color{blue}{\frac{-2 \cdot {c}^{3}}{{b}^{4}}} - {\left(\frac{c}{b}\right)}^{2}, -c\right)}{b} \]

   13. Simplified 93.2%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(a, a \cdot \frac{-2 \cdot {c}^{3}}{{b}^{4}} - {\left(\frac{c}{b}\right)}^{2}, -c\right)}}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(-4 \cdot a + \frac{{b}^{2}}{c}\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, a \cdot \frac{-2 \cdot {c}^{3}}{{b}^{4}} - {\left(\frac{c}{b}\right)}^{2}, -c\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.1:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(-4 \cdot a + \frac{{b}^{2}}{c}\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(-2 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} - \frac{a}{{b}^{3}}\right) + \frac{-1}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.1)
   (/ (- (sqrt (* c (+ (* -4.0 a) (/ (pow b 2.0) c)))) b) (* 2.0 a))
   (*
    c
    (+
     (* c (- (* -2.0 (/ (* c (pow a 2.0)) (pow b 5.0))) (/ a (pow b 3.0))))
     (/ -1.0 b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.1) {
		tmp = (sqrt((c * ((-4.0 * a) + (pow(b, 2.0) / c)))) - b) / (2.0 * a);
	} else {
		tmp = c * ((c * ((-2.0 * ((c * pow(a, 2.0)) / pow(b, 5.0))) - (a / pow(b, 3.0)))) + (-1.0 / b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 2.1d0) then
        tmp = (sqrt((c * (((-4.0d0) * a) + ((b ** 2.0d0) / c)))) - b) / (2.0d0 * a)
    else
        tmp = c * ((c * (((-2.0d0) * ((c * (a ** 2.0d0)) / (b ** 5.0d0))) - (a / (b ** 3.0d0)))) + ((-1.0d0) / b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.1) {
		tmp = (Math.sqrt((c * ((-4.0 * a) + (Math.pow(b, 2.0) / c)))) - b) / (2.0 * a);
	} else {
		tmp = c * ((c * ((-2.0 * ((c * Math.pow(a, 2.0)) / Math.pow(b, 5.0))) - (a / Math.pow(b, 3.0)))) + (-1.0 / b));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.1:
		tmp = (math.sqrt((c * ((-4.0 * a) + (math.pow(b, 2.0) / c)))) - b) / (2.0 * a)
	else:
		tmp = c * ((c * ((-2.0 * ((c * math.pow(a, 2.0)) / math.pow(b, 5.0))) - (a / math.pow(b, 3.0)))) + (-1.0 / b))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.1)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(Float64(-4.0 * a) + Float64((b ^ 2.0) / c)))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(c * Float64(Float64(c * Float64(Float64(-2.0 * Float64(Float64(c * (a ^ 2.0)) / (b ^ 5.0))) - Float64(a / (b ^ 3.0)))) + Float64(-1.0 / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.1)
		tmp = (sqrt((c * ((-4.0 * a) + ((b ^ 2.0) / c)))) - b) / (2.0 * a);
	else
		tmp = c * ((c * ((-2.0 * ((c * (a ^ 2.0)) / (b ^ 5.0))) - (a / (b ^ 3.0)))) + (-1.0 / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.1], N[(N[(N[Sqrt[N[(c * N[(N[(-4.0 * a), $MachinePrecision] + N[(N[Power[b, 2.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(c * N[(N[(-2.0 * N[(N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.10000000000000009

   1. Initial program 86.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 86.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

      2. +-commutative 86.0%

         \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]

      3. sqr-neg 86.0%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{a \cdot 2} \]

      4. unsub-neg 86.0%

         \[\leadsto \frac{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right) - \left(4 \cdot a\right) \cdot c} - b}}{a \cdot 2} \]

      5. sqr-neg 86.0%

         \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b} - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \]

      6. fma-neg 86.0%

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}} - b}{a \cdot 2} \]

      7. distribute-lft-neg-in 86.0%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-4 \cdot a\right) \cdot c}\right)} - b}{a \cdot 2} \]

      8. *-commutative 86.0%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-4 \cdot a\right)}\right)} - b}{a \cdot 2} \]

      9. *-commutative 86.0%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-\color{blue}{a \cdot 4}\right)\right)} - b}{a \cdot 2} \]

      10. distribute-rgt-neg-in 86.0%

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(a \cdot \left(-4\right)\right)}\right)} - b}{a \cdot 2} \]

      11. metadata-eval 86.0%

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot \color{blue}{-4}\right)\right)} - b}{a \cdot 2} \]

   3. Simplified 86.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in c around inf 86.0%

      \[\leadsto \frac{\sqrt{\color{blue}{c \cdot \left(-4 \cdot a + \frac{{b}^{2}}{c}\right)}} - b}{a \cdot 2} \]

   if 2.10000000000000009 < b

   1. Initial program 45.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 45.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

      2. +-commutative 45.8%

         \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]

      3. sqr-neg 45.8%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{a \cdot 2} \]

      4. unsub-neg 45.8%

         \[\leadsto \frac{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right) - \left(4 \cdot a\right) \cdot c} - b}}{a \cdot 2} \]

      5. sqr-neg 45.8%

         \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b} - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \]

      6. fma-neg 45.9%

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}} - b}{a \cdot 2} \]

      7. distribute-lft-neg-in 45.9%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-4 \cdot a\right) \cdot c}\right)} - b}{a \cdot 2} \]

      8. *-commutative 45.9%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-4 \cdot a\right)}\right)} - b}{a \cdot 2} \]

      9. *-commutative 45.9%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-\color{blue}{a \cdot 4}\right)\right)} - b}{a \cdot 2} \]

      10. distribute-rgt-neg-in 45.9%

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(a \cdot \left(-4\right)\right)}\right)} - b}{a \cdot 2} \]

      11. metadata-eval 45.9%

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot \color{blue}{-4}\right)\right)} - b}{a \cdot 2} \]

   3. Simplified 45.9%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in c around 0 93.0%

      \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -1 \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{b}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.1:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(-4 \cdot a + \frac{{b}^{2}}{c}\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(-2 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} - \frac{a}{{b}^{3}}\right) + \frac{-1}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(-4 \cdot a + \frac{{b}^{2}}{c}\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(-1 + c \cdot \left(a \cdot \left(-2 \cdot \frac{c \cdot a}{{b}^{4}} + \frac{-1}{{b}^{2}}\right)\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.0)
   (/ (- (sqrt (* c (+ (* -4.0 a) (/ (pow b 2.0) c)))) b) (* 2.0 a))
   (/
    (*
     c
     (+
      -1.0
      (* c (* a (+ (* -2.0 (/ (* c a) (pow b 4.0))) (/ -1.0 (pow b 2.0)))))))
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.0) {
		tmp = (sqrt((c * ((-4.0 * a) + (pow(b, 2.0) / c)))) - b) / (2.0 * a);
	} else {
		tmp = (c * (-1.0 + (c * (a * ((-2.0 * ((c * a) / pow(b, 4.0))) + (-1.0 / pow(b, 2.0))))))) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 2.0d0) then
        tmp = (sqrt((c * (((-4.0d0) * a) + ((b ** 2.0d0) / c)))) - b) / (2.0d0 * a)
    else
        tmp = (c * ((-1.0d0) + (c * (a * (((-2.0d0) * ((c * a) / (b ** 4.0d0))) + ((-1.0d0) / (b ** 2.0d0))))))) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.0) {
		tmp = (Math.sqrt((c * ((-4.0 * a) + (Math.pow(b, 2.0) / c)))) - b) / (2.0 * a);
	} else {
		tmp = (c * (-1.0 + (c * (a * ((-2.0 * ((c * a) / Math.pow(b, 4.0))) + (-1.0 / Math.pow(b, 2.0))))))) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.0:
		tmp = (math.sqrt((c * ((-4.0 * a) + (math.pow(b, 2.0) / c)))) - b) / (2.0 * a)
	else:
		tmp = (c * (-1.0 + (c * (a * ((-2.0 * ((c * a) / math.pow(b, 4.0))) + (-1.0 / math.pow(b, 2.0))))))) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.0)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(Float64(-4.0 * a) + Float64((b ^ 2.0) / c)))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(c * Float64(-1.0 + Float64(c * Float64(a * Float64(Float64(-2.0 * Float64(Float64(c * a) / (b ^ 4.0))) + Float64(-1.0 / (b ^ 2.0))))))) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.0)
		tmp = (sqrt((c * ((-4.0 * a) + ((b ^ 2.0) / c)))) - b) / (2.0 * a);
	else
		tmp = (c * (-1.0 + (c * (a * ((-2.0 * ((c * a) / (b ^ 4.0))) + (-1.0 / (b ^ 2.0))))))) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.0], N[(N[(N[Sqrt[N[(c * N[(N[(-4.0 * a), $MachinePrecision] + N[(N[Power[b, 2.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(-1.0 + N[(c * N[(a * N[(N[(-2.0 * N[(N[(c * a), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2

   1. Initial program 86.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 86.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

      2. +-commutative 86.0%

         \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]

      3. sqr-neg 86.0%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{a \cdot 2} \]

      4. unsub-neg 86.0%

         \[\leadsto \frac{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right) - \left(4 \cdot a\right) \cdot c} - b}}{a \cdot 2} \]

      5. sqr-neg 86.0%

         \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b} - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \]

      6. fma-neg 86.0%

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}} - b}{a \cdot 2} \]

      7. distribute-lft-neg-in 86.0%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-4 \cdot a\right) \cdot c}\right)} - b}{a \cdot 2} \]

      8. *-commutative 86.0%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-4 \cdot a\right)}\right)} - b}{a \cdot 2} \]

      9. *-commutative 86.0%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-\color{blue}{a \cdot 4}\right)\right)} - b}{a \cdot 2} \]

      10. distribute-rgt-neg-in 86.0%

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(a \cdot \left(-4\right)\right)}\right)} - b}{a \cdot 2} \]

      11. metadata-eval 86.0%

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot \color{blue}{-4}\right)\right)} - b}{a \cdot 2} \]

   3. Simplified 86.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in c around inf 86.0%

      \[\leadsto \frac{\sqrt{\color{blue}{c \cdot \left(-4 \cdot a + \frac{{b}^{2}}{c}\right)}} - b}{a \cdot 2} \]

   if 2 < b

   1. Initial program 45.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 45.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

      2. +-commutative 45.8%

         \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]

      3. sqr-neg 45.8%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{a \cdot 2} \]

      4. unsub-neg 45.8%

         \[\leadsto \frac{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right) - \left(4 \cdot a\right) \cdot c} - b}}{a \cdot 2} \]

      5. sqr-neg 45.8%

         \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b} - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \]

      6. fma-neg 45.9%

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}} - b}{a \cdot 2} \]

      7. distribute-lft-neg-in 45.9%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-4 \cdot a\right) \cdot c}\right)} - b}{a \cdot 2} \]

      8. *-commutative 45.9%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-4 \cdot a\right)}\right)} - b}{a \cdot 2} \]

      9. *-commutative 45.9%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-\color{blue}{a \cdot 4}\right)\right)} - b}{a \cdot 2} \]

      10. distribute-rgt-neg-in 45.9%

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(a \cdot \left(-4\right)\right)}\right)} - b}{a \cdot 2} \]

      11. metadata-eval 45.9%

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot \color{blue}{-4}\right)\right)} - b}{a \cdot 2} \]

   3. Simplified 45.9%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 93.2%

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 61.5%

         \[\leadsto \frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}}{b} \]

      2. expm1-undefine 51.9%

         \[\leadsto \frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \color{blue}{\left(e^{\mathsf{log1p}\left(-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)} - 1\right)}}{b} \]

      3. neg-mul-1 51.9%

         \[\leadsto \frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(e^{\mathsf{log1p}\left(\color{blue}{\left(-c\right)} + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)} - 1\right)}{b} \]

      4. +-commutative 51.9%

         \[\leadsto \frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(e^{\mathsf{log1p}\left(\color{blue}{-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \left(-c\right)}\right)} - 1\right)}{b} \]

      5. fma-define 51.9%

         \[\leadsto \frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(-1, \frac{a \cdot {c}^{2}}{{b}^{2}}, -c\right)}\right)} - 1\right)}{b} \]

      6. associate-/l* 51.9%

         \[\leadsto \frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(-1, \color{blue}{a \cdot \frac{{c}^{2}}{{b}^{2}}}, -c\right)\right)} - 1\right)}{b} \]

   7. Applied egg-rr 51.9%

      \[\leadsto \frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \color{blue}{\left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(-1, a \cdot \frac{{c}^{2}}{{b}^{2}}, -c\right)\right)} - 1\right)}}{b} \]
   8. Step-by-step derivation

      1. sub-neg 51.9%

         \[\leadsto \frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \color{blue}{\left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(-1, a \cdot \frac{{c}^{2}}{{b}^{2}}, -c\right)\right)} + \left(-1\right)\right)}}{b} \]

      2. metadata-eval 51.9%

         \[\leadsto \frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(-1, a \cdot \frac{{c}^{2}}{{b}^{2}}, -c\right)\right)} + \color{blue}{-1}\right)}{b} \]

      3. +-commutative 51.9%

         \[\leadsto \frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \color{blue}{\left(-1 + e^{\mathsf{log1p}\left(\mathsf{fma}\left(-1, a \cdot \frac{{c}^{2}}{{b}^{2}}, -c\right)\right)}\right)}}{b} \]

      4. log1p-undefine 51.9%

         \[\leadsto \frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 + e^{\color{blue}{\log \left(1 + \mathsf{fma}\left(-1, a \cdot \frac{{c}^{2}}{{b}^{2}}, -c\right)\right)}}\right)}{b} \]

      5. rem-exp-log 83.5%

         \[\leadsto \frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 + \color{blue}{\left(1 + \mathsf{fma}\left(-1, a \cdot \frac{{c}^{2}}{{b}^{2}}, -c\right)\right)}\right)}{b} \]

      6. fma-undefine 83.5%

         \[\leadsto \frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 + \left(1 + \color{blue}{\left(-1 \cdot \left(a \cdot \frac{{c}^{2}}{{b}^{2}}\right) + \left(-c\right)\right)}\right)\right)}{b} \]

      7. neg-mul-1 83.5%

         \[\leadsto \frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 + \left(1 + \left(\color{blue}{\left(-a \cdot \frac{{c}^{2}}{{b}^{2}}\right)} + \left(-c\right)\right)\right)\right)}{b} \]

      8. distribute-neg-in 83.5%

         \[\leadsto \frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 + \left(1 + \color{blue}{\left(-\left(a \cdot \frac{{c}^{2}}{{b}^{2}} + c\right)\right)}\right)\right)}{b} \]

      9. +-commutative 83.5%

         \[\leadsto \frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 + \left(1 + \left(-\color{blue}{\left(c + a \cdot \frac{{c}^{2}}{{b}^{2}}\right)}\right)\right)\right)}{b} \]

      10. unsub-neg 83.5%

          \[\leadsto \frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 + \color{blue}{\left(1 - \left(c + a \cdot \frac{{c}^{2}}{{b}^{2}}\right)\right)}\right)}{b} \]

      11. +-commutative 83.5%

          \[\leadsto \frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 + \left(1 - \color{blue}{\left(a \cdot \frac{{c}^{2}}{{b}^{2}} + c\right)}\right)\right)}{b} \]

      12. fma-define 83.5%

          \[\leadsto \frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 + \left(1 - \color{blue}{\mathsf{fma}\left(a, \frac{{c}^{2}}{{b}^{2}}, c\right)}\right)\right)}{b} \]

   9. Simplified 83.5%

      \[\leadsto \frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \color{blue}{\left(-1 + \left(1 - \mathsf{fma}\left(a, {\left(\frac{c}{-b}\right)}^{2}, c\right)\right)\right)}}{b} \]

   10. Taylor expanded in c around 0 93.0%

       \[\leadsto \frac{\color{blue}{c \cdot \left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} - \frac{a}{{b}^{2}}\right) - 1\right)}}{b} \]

   11. Taylor expanded in a around 0 93.0%

       \[\leadsto \frac{c \cdot \left(c \cdot \color{blue}{\left(a \cdot \left(-2 \cdot \frac{a \cdot c}{{b}^{4}} - \frac{1}{{b}^{2}}\right)\right)} - 1\right)}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(-4 \cdot a + \frac{{b}^{2}}{c}\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(-1 + c \cdot \left(a \cdot \left(-2 \cdot \frac{c \cdot a}{{b}^{4}} + \frac{-1}{{b}^{2}}\right)\right)\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.6:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + a \cdot {\left(\frac{c}{-b}\right)}^{2}}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.6)
   (/ (- (sqrt (fma b b (* c (* -4.0 a)))) b) (* 2.0 a))
   (/ (+ c (* a (pow (/ c (- b)) 2.0))) (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.6) {
		tmp = (sqrt(fma(b, b, (c * (-4.0 * a)))) - b) / (2.0 * a);
	} else {
		tmp = (c + (a * pow((c / -b), 2.0))) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.6)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(-4.0 * a)))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(c + Float64(a * (Float64(c / Float64(-b)) ^ 2.0))) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.6], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(-4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c + N[(a * N[Power[N[(c / (-b)), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.5999999999999996

   1. Initial program 84.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 84.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

      2. +-commutative 84.1%

         \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]

      3. sqr-neg 84.1%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{a \cdot 2} \]

      4. unsub-neg 84.1%

         \[\leadsto \frac{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right) - \left(4 \cdot a\right) \cdot c} - b}}{a \cdot 2} \]

      5. sqr-neg 84.1%

         \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b} - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \]

      6. fma-neg 84.1%

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}} - b}{a \cdot 2} \]

      7. distribute-lft-neg-in 84.1%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-4 \cdot a\right) \cdot c}\right)} - b}{a \cdot 2} \]

      8. *-commutative 84.1%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-4 \cdot a\right)}\right)} - b}{a \cdot 2} \]

      9. *-commutative 84.1%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-\color{blue}{a \cdot 4}\right)\right)} - b}{a \cdot 2} \]

      10. distribute-rgt-neg-in 84.1%

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(a \cdot \left(-4\right)\right)}\right)} - b}{a \cdot 2} \]

      11. metadata-eval 84.1%

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot \color{blue}{-4}\right)\right)} - b}{a \cdot 2} \]

   3. Simplified 84.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}} \]
   4. Add Preprocessing

   if 5.5999999999999996 < b

   1. Initial program 44.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 44.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

      2. +-commutative 44.3%

         \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]

      3. sqr-neg 44.3%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{a \cdot 2} \]

      4. unsub-neg 44.3%

         \[\leadsto \frac{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right) - \left(4 \cdot a\right) \cdot c} - b}}{a \cdot 2} \]

      5. sqr-neg 44.3%

         \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b} - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \]

      6. fma-neg 44.4%

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}} - b}{a \cdot 2} \]

      7. distribute-lft-neg-in 44.4%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-4 \cdot a\right) \cdot c}\right)} - b}{a \cdot 2} \]

      8. *-commutative 44.4%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-4 \cdot a\right)}\right)} - b}{a \cdot 2} \]

      9. *-commutative 44.4%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-\color{blue}{a \cdot 4}\right)\right)} - b}{a \cdot 2} \]

      10. distribute-rgt-neg-in 44.4%

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(a \cdot \left(-4\right)\right)}\right)} - b}{a \cdot 2} \]

      11. metadata-eval 44.4%

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot \color{blue}{-4}\right)\right)} - b}{a \cdot 2} \]

   3. Simplified 44.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 89.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]
   6. Step-by-step derivation

      1. mul-1-neg 89.7%

         \[\leadsto -1 \cdot \frac{c}{b} + \color{blue}{\left(-\frac{a \cdot {c}^{2}}{{b}^{3}}\right)} \]

      2. unsub-neg 89.7%

         \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} - \frac{a \cdot {c}^{2}}{{b}^{3}}} \]

      3. mul-1-neg 89.7%

         \[\leadsto \color{blue}{\left(-\frac{c}{b}\right)} - \frac{a \cdot {c}^{2}}{{b}^{3}} \]

      4. distribute-neg-frac2 89.7%

         \[\leadsto \color{blue}{\frac{c}{-b}} - \frac{a \cdot {c}^{2}}{{b}^{3}} \]

      5. associate-/l* 89.7%

         \[\leadsto \frac{c}{-b} - \color{blue}{a \cdot \frac{{c}^{2}}{{b}^{3}}} \]

   7. Simplified 89.7%

      \[\leadsto \color{blue}{\frac{c}{-b} - a \cdot \frac{{c}^{2}}{{b}^{3}}} \]

   8. Taylor expanded in b around inf 89.7%

      \[\leadsto \color{blue}{\frac{-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
   9. Step-by-step derivation

      1. distribute-lft-out 89.7%

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(c + \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}}{b} \]

      2. associate-*r/ 89.7%

         \[\leadsto \color{blue}{-1 \cdot \frac{c + \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]

      3. mul-1-neg 89.7%

         \[\leadsto \color{blue}{-\frac{c + \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]

      4. associate-*r/ 89.7%

         \[\leadsto -\frac{c + \color{blue}{a \cdot \frac{{c}^{2}}{{b}^{2}}}}{b} \]

      5. distribute-neg-frac2 89.7%

         \[\leadsto \color{blue}{\frac{c + a \cdot \frac{{c}^{2}}{{b}^{2}}}{-b}} \]

      6. +-commutative 89.7%

         \[\leadsto \frac{\color{blue}{a \cdot \frac{{c}^{2}}{{b}^{2}} + c}}{-b} \]

      7. fma-define 89.7%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(a, \frac{{c}^{2}}{{b}^{2}}, c\right)}}{-b} \]

      8. unpow2 89.7%

         \[\leadsto \frac{\mathsf{fma}\left(a, \frac{\color{blue}{c \cdot c}}{{b}^{2}}, c\right)}{-b} \]

      9. unpow2 89.7%

         \[\leadsto \frac{\mathsf{fma}\left(a, \frac{c \cdot c}{\color{blue}{b \cdot b}}, c\right)}{-b} \]

      10. times-frac 89.7%

          \[\leadsto \frac{\mathsf{fma}\left(a, \color{blue}{\frac{c}{b} \cdot \frac{c}{b}}, c\right)}{-b} \]

      11. sqr-neg 89.7%

          \[\leadsto \frac{\mathsf{fma}\left(a, \color{blue}{\left(-\frac{c}{b}\right) \cdot \left(-\frac{c}{b}\right)}, c\right)}{-b} \]

      12. distribute-frac-neg2 89.7%

          \[\leadsto \frac{\mathsf{fma}\left(a, \color{blue}{\frac{c}{-b}} \cdot \left(-\frac{c}{b}\right), c\right)}{-b} \]

      13. distribute-frac-neg2 89.7%

          \[\leadsto \frac{\mathsf{fma}\left(a, \frac{c}{-b} \cdot \color{blue}{\frac{c}{-b}}, c\right)}{-b} \]

      14. unpow2 89.7%

          \[\leadsto \frac{\mathsf{fma}\left(a, \color{blue}{{\left(\frac{c}{-b}\right)}^{2}}, c\right)}{-b} \]

   10. Simplified 89.7%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, {\left(\frac{c}{-b}\right)}^{2}, c\right)}{-b}} \]
   11. Step-by-step derivation

       1. fma-undefine 89.7%

          \[\leadsto \frac{\color{blue}{a \cdot {\left(\frac{c}{-b}\right)}^{2} + c}}{-b} \]

   12. Applied egg-rr 89.7%

       \[\leadsto \frac{\color{blue}{a \cdot {\left(\frac{c}{-b}\right)}^{2} + c}}{-b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.6:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + a \cdot {\left(\frac{c}{-b}\right)}^{2}}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.6:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + a \cdot {\left(\frac{c}{-b}\right)}^{2}}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.6)
   (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* 2.0 a))
   (/ (+ c (* a (pow (/ c (- b)) 2.0))) (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.6) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
	} else {
		tmp = (c + (a * pow((c / -b), 2.0))) / -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 5.6d0) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (2.0d0 * a)
    else
        tmp = (c + (a * ((c / -b) ** 2.0d0))) / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.6) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
	} else {
		tmp = (c + (a * Math.pow((c / -b), 2.0))) / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.6:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a)
	else:
		tmp = (c + (a * math.pow((c / -b), 2.0))) / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.6)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(c + Float64(a * (Float64(c / Float64(-b)) ^ 2.0))) / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 5.6)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
	else
		tmp = (c + (a * ((c / -b) ^ 2.0))) / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.6], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c + N[(a * N[Power[N[(c / (-b)), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.5999999999999996

   1. Initial program 84.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   if 5.5999999999999996 < b

   1. Initial program 44.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 44.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

      2. +-commutative 44.3%

         \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]

      3. sqr-neg 44.3%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{a \cdot 2} \]

      4. unsub-neg 44.3%

         \[\leadsto \frac{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right) - \left(4 \cdot a\right) \cdot c} - b}}{a \cdot 2} \]

      5. sqr-neg 44.3%

         \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b} - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \]

      6. fma-neg 44.4%

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}} - b}{a \cdot 2} \]

      7. distribute-lft-neg-in 44.4%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-4 \cdot a\right) \cdot c}\right)} - b}{a \cdot 2} \]

      8. *-commutative 44.4%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-4 \cdot a\right)}\right)} - b}{a \cdot 2} \]

      9. *-commutative 44.4%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-\color{blue}{a \cdot 4}\right)\right)} - b}{a \cdot 2} \]

      10. distribute-rgt-neg-in 44.4%

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(a \cdot \left(-4\right)\right)}\right)} - b}{a \cdot 2} \]

      11. metadata-eval 44.4%

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot \color{blue}{-4}\right)\right)} - b}{a \cdot 2} \]

   3. Simplified 44.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 89.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]
   6. Step-by-step derivation

      1. mul-1-neg 89.7%

         \[\leadsto -1 \cdot \frac{c}{b} + \color{blue}{\left(-\frac{a \cdot {c}^{2}}{{b}^{3}}\right)} \]

      2. unsub-neg 89.7%

         \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} - \frac{a \cdot {c}^{2}}{{b}^{3}}} \]

      3. mul-1-neg 89.7%

         \[\leadsto \color{blue}{\left(-\frac{c}{b}\right)} - \frac{a \cdot {c}^{2}}{{b}^{3}} \]

      4. distribute-neg-frac2 89.7%

         \[\leadsto \color{blue}{\frac{c}{-b}} - \frac{a \cdot {c}^{2}}{{b}^{3}} \]

      5. associate-/l* 89.7%

         \[\leadsto \frac{c}{-b} - \color{blue}{a \cdot \frac{{c}^{2}}{{b}^{3}}} \]

   7. Simplified 89.7%

      \[\leadsto \color{blue}{\frac{c}{-b} - a \cdot \frac{{c}^{2}}{{b}^{3}}} \]

   8. Taylor expanded in b around inf 89.7%

      \[\leadsto \color{blue}{\frac{-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
   9. Step-by-step derivation

      1. distribute-lft-out 89.7%

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(c + \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}}{b} \]

      2. associate-*r/ 89.7%

         \[\leadsto \color{blue}{-1 \cdot \frac{c + \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]

      3. mul-1-neg 89.7%

         \[\leadsto \color{blue}{-\frac{c + \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]

      4. associate-*r/ 89.7%

         \[\leadsto -\frac{c + \color{blue}{a \cdot \frac{{c}^{2}}{{b}^{2}}}}{b} \]

      5. distribute-neg-frac2 89.7%

         \[\leadsto \color{blue}{\frac{c + a \cdot \frac{{c}^{2}}{{b}^{2}}}{-b}} \]

      6. +-commutative 89.7%

         \[\leadsto \frac{\color{blue}{a \cdot \frac{{c}^{2}}{{b}^{2}} + c}}{-b} \]

      7. fma-define 89.7%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(a, \frac{{c}^{2}}{{b}^{2}}, c\right)}}{-b} \]

      8. unpow2 89.7%

         \[\leadsto \frac{\mathsf{fma}\left(a, \frac{\color{blue}{c \cdot c}}{{b}^{2}}, c\right)}{-b} \]

      9. unpow2 89.7%

         \[\leadsto \frac{\mathsf{fma}\left(a, \frac{c \cdot c}{\color{blue}{b \cdot b}}, c\right)}{-b} \]

      10. times-frac 89.7%

          \[\leadsto \frac{\mathsf{fma}\left(a, \color{blue}{\frac{c}{b} \cdot \frac{c}{b}}, c\right)}{-b} \]

      11. sqr-neg 89.7%

          \[\leadsto \frac{\mathsf{fma}\left(a, \color{blue}{\left(-\frac{c}{b}\right) \cdot \left(-\frac{c}{b}\right)}, c\right)}{-b} \]

      12. distribute-frac-neg2 89.7%

          \[\leadsto \frac{\mathsf{fma}\left(a, \color{blue}{\frac{c}{-b}} \cdot \left(-\frac{c}{b}\right), c\right)}{-b} \]

      13. distribute-frac-neg2 89.7%

          \[\leadsto \frac{\mathsf{fma}\left(a, \frac{c}{-b} \cdot \color{blue}{\frac{c}{-b}}, c\right)}{-b} \]

      14. unpow2 89.7%

          \[\leadsto \frac{\mathsf{fma}\left(a, \color{blue}{{\left(\frac{c}{-b}\right)}^{2}}, c\right)}{-b} \]

   10. Simplified 89.7%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, {\left(\frac{c}{-b}\right)}^{2}, c\right)}{-b}} \]
   11. Step-by-step derivation

       1. fma-undefine 89.7%

          \[\leadsto \frac{\color{blue}{a \cdot {\left(\frac{c}{-b}\right)}^{2} + c}}{-b} \]

   12. Applied egg-rr 89.7%

       \[\leadsto \frac{\color{blue}{a \cdot {\left(\frac{c}{-b}\right)}^{2} + c}}{-b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.6:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + a \cdot {\left(\frac{c}{-b}\right)}^{2}}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{c + a \cdot {\left(\frac{c}{-b}\right)}^{2}}{-b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ c (* a (pow (/ c (- b)) 2.0))) (- b)))

C

double code(double a, double b, double c) {
	return (c + (a * pow((c / -b), 2.0))) / -b;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (c + (a * ((c / -b) ** 2.0d0))) / -b
end function

Java

public static double code(double a, double b, double c) {
	return (c + (a * Math.pow((c / -b), 2.0))) / -b;
}

Python

def code(a, b, c):
	return (c + (a * math.pow((c / -b), 2.0))) / -b

Julia

function code(a, b, c)
	return Float64(Float64(c + Float64(a * (Float64(c / Float64(-b)) ^ 2.0))) / Float64(-b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (c + (a * ((c / -b) ^ 2.0))) / -b;
end

Wolfram

code[a_, b_, c_] := N[(N[(c + N[(a * N[Power[N[(c / (-b)), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-b)), $MachinePrecision]

Derivation

1. Initial program 53.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 53.3%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   2. +-commutative 53.3%

      \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]

   3. sqr-neg 53.3%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{a \cdot 2} \]

   4. unsub-neg 53.3%

      \[\leadsto \frac{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right) - \left(4 \cdot a\right) \cdot c} - b}}{a \cdot 2} \]

   5. sqr-neg 53.3%

      \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b} - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \]

   6. fma-neg 53.4%

      \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}} - b}{a \cdot 2} \]

   7. distribute-lft-neg-in 53.4%

      \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-4 \cdot a\right) \cdot c}\right)} - b}{a \cdot 2} \]

   8. *-commutative 53.4%

      \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-4 \cdot a\right)}\right)} - b}{a \cdot 2} \]

   9. *-commutative 53.4%

      \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-\color{blue}{a \cdot 4}\right)\right)} - b}{a \cdot 2} \]

   10. distribute-rgt-neg-in 53.4%

       \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(a \cdot \left(-4\right)\right)}\right)} - b}{a \cdot 2} \]

   11. metadata-eval 53.4%

       \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot \color{blue}{-4}\right)\right)} - b}{a \cdot 2} \]

3. Simplified 53.4%

   \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in a around 0 81.9%

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]
6. Step-by-step derivation

   1. mul-1-neg 81.9%

      \[\leadsto -1 \cdot \frac{c}{b} + \color{blue}{\left(-\frac{a \cdot {c}^{2}}{{b}^{3}}\right)} \]

   2. unsub-neg 81.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} - \frac{a \cdot {c}^{2}}{{b}^{3}}} \]

   3. mul-1-neg 81.9%

      \[\leadsto \color{blue}{\left(-\frac{c}{b}\right)} - \frac{a \cdot {c}^{2}}{{b}^{3}} \]

   4. distribute-neg-frac2 81.9%

      \[\leadsto \color{blue}{\frac{c}{-b}} - \frac{a \cdot {c}^{2}}{{b}^{3}} \]

   5. associate-/l* 81.9%

      \[\leadsto \frac{c}{-b} - \color{blue}{a \cdot \frac{{c}^{2}}{{b}^{3}}} \]

7. Simplified 81.9%

   \[\leadsto \color{blue}{\frac{c}{-b} - a \cdot \frac{{c}^{2}}{{b}^{3}}} \]

8. Taylor expanded in b around inf 81.9%

   \[\leadsto \color{blue}{\frac{-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
9. Step-by-step derivation

   1. distribute-lft-out 81.9%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(c + \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}}{b} \]

   2. associate-*r/ 81.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{c + \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]

   3. mul-1-neg 81.9%

      \[\leadsto \color{blue}{-\frac{c + \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]

   4. associate-*r/ 81.9%

      \[\leadsto -\frac{c + \color{blue}{a \cdot \frac{{c}^{2}}{{b}^{2}}}}{b} \]

   5. distribute-neg-frac2 81.9%

      \[\leadsto \color{blue}{\frac{c + a \cdot \frac{{c}^{2}}{{b}^{2}}}{-b}} \]

   6. +-commutative 81.9%

      \[\leadsto \frac{\color{blue}{a \cdot \frac{{c}^{2}}{{b}^{2}} + c}}{-b} \]

   7. fma-define 81.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(a, \frac{{c}^{2}}{{b}^{2}}, c\right)}}{-b} \]

   8. unpow2 81.9%

      \[\leadsto \frac{\mathsf{fma}\left(a, \frac{\color{blue}{c \cdot c}}{{b}^{2}}, c\right)}{-b} \]

   9. unpow2 81.9%

      \[\leadsto \frac{\mathsf{fma}\left(a, \frac{c \cdot c}{\color{blue}{b \cdot b}}, c\right)}{-b} \]

   10. times-frac 81.9%

       \[\leadsto \frac{\mathsf{fma}\left(a, \color{blue}{\frac{c}{b} \cdot \frac{c}{b}}, c\right)}{-b} \]

   11. sqr-neg 81.9%

       \[\leadsto \frac{\mathsf{fma}\left(a, \color{blue}{\left(-\frac{c}{b}\right) \cdot \left(-\frac{c}{b}\right)}, c\right)}{-b} \]

   12. distribute-frac-neg2 81.9%

       \[\leadsto \frac{\mathsf{fma}\left(a, \color{blue}{\frac{c}{-b}} \cdot \left(-\frac{c}{b}\right), c\right)}{-b} \]

   13. distribute-frac-neg2 81.9%

       \[\leadsto \frac{\mathsf{fma}\left(a, \frac{c}{-b} \cdot \color{blue}{\frac{c}{-b}}, c\right)}{-b} \]

   14. unpow2 81.9%

       \[\leadsto \frac{\mathsf{fma}\left(a, \color{blue}{{\left(\frac{c}{-b}\right)}^{2}}, c\right)}{-b} \]

10. Simplified 81.9%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, {\left(\frac{c}{-b}\right)}^{2}, c\right)}{-b}} \]
11. Step-by-step derivation

    1. fma-undefine 81.9%

       \[\leadsto \frac{\color{blue}{a \cdot {\left(\frac{c}{-b}\right)}^{2} + c}}{-b} \]

12. Applied egg-rr 81.9%

    \[\leadsto \frac{\color{blue}{a \cdot {\left(\frac{c}{-b}\right)}^{2} + c}}{-b} \]

13. Final simplification 81.9%

    \[\leadsto \frac{c + a \cdot {\left(\frac{c}{-b}\right)}^{2}}{-b} \]
14. Add Preprocessing

Alternative 8: 64.4% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ \frac{c}{-b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ c (- b)))

C

double code(double a, double b, double c) {
	return c / -b;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c / -b
end function

Java

public static double code(double a, double b, double c) {
	return c / -b;
}

Python

def code(a, b, c):
	return c / -b

Julia

function code(a, b, c)
	return Float64(c / Float64(-b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c / -b;
end

Wolfram

code[a_, b_, c_] := N[(c / (-b)), $MachinePrecision]

Derivation

1. Initial program 53.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 53.3%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   2. +-commutative 53.3%

      \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]

   3. sqr-neg 53.3%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{a \cdot 2} \]

   4. unsub-neg 53.3%

      \[\leadsto \frac{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right) - \left(4 \cdot a\right) \cdot c} - b}}{a \cdot 2} \]

   5. sqr-neg 53.3%

      \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b} - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \]

   6. fma-neg 53.4%

      \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}} - b}{a \cdot 2} \]

   7. distribute-lft-neg-in 53.4%

      \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-4 \cdot a\right) \cdot c}\right)} - b}{a \cdot 2} \]

   8. *-commutative 53.4%

      \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-4 \cdot a\right)}\right)} - b}{a \cdot 2} \]

   9. *-commutative 53.4%

      \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-\color{blue}{a \cdot 4}\right)\right)} - b}{a \cdot 2} \]

   10. distribute-rgt-neg-in 53.4%

       \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(a \cdot \left(-4\right)\right)}\right)} - b}{a \cdot 2} \]

   11. metadata-eval 53.4%

       \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot \color{blue}{-4}\right)\right)} - b}{a \cdot 2} \]

3. Simplified 53.4%

   \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in b around inf 66.0%

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
6. Step-by-step derivation

   1. associate-*r/ 66.0%

      \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} \]

   2. mul-1-neg 66.0%

      \[\leadsto \frac{\color{blue}{-c}}{b} \]

7. Simplified 66.0%

   \[\leadsto \color{blue}{\frac{-c}{b}} \]

8. Final simplification 66.0%

   \[\leadsto \frac{c}{-b} \]
9. Add Preprocessing

Alternative 9: 3.2% accurate, 116.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 0.0)

C

double code(double a, double b, double c) {
	return 0.0;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0
end function

Java

public static double code(double a, double b, double c) {
	return 0.0;
}

Python

def code(a, b, c):
	return 0.0

Julia

function code(a, b, c)
	return 0.0
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0;
end

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 53.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 53.3%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   2. +-commutative 53.3%

      \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]

   3. sqr-neg 53.3%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{a \cdot 2} \]

   4. unsub-neg 53.3%

      \[\leadsto \frac{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right) - \left(4 \cdot a\right) \cdot c} - b}}{a \cdot 2} \]

   5. sqr-neg 53.3%

      \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b} - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \]

   6. fma-neg 53.4%

      \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}} - b}{a \cdot 2} \]

   7. distribute-lft-neg-in 53.4%

      \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-4 \cdot a\right) \cdot c}\right)} - b}{a \cdot 2} \]

   8. *-commutative 53.4%

      \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-4 \cdot a\right)}\right)} - b}{a \cdot 2} \]

   9. *-commutative 53.4%

      \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-\color{blue}{a \cdot 4}\right)\right)} - b}{a \cdot 2} \]

   10. distribute-rgt-neg-in 53.4%

       \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(a \cdot \left(-4\right)\right)}\right)} - b}{a \cdot 2} \]

   11. metadata-eval 53.4%

       \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot \color{blue}{-4}\right)\right)} - b}{a \cdot 2} \]

3. Simplified 53.4%

   \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in b around inf 66.0%

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
6. Step-by-step derivation

   1. associate-*r/ 66.0%

      \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} \]

   2. mul-1-neg 66.0%

      \[\leadsto \frac{\color{blue}{-c}}{b} \]

7. Simplified 66.0%

   \[\leadsto \color{blue}{\frac{-c}{b}} \]
8. Step-by-step derivation

   1. distribute-frac-neg 66.0%

      \[\leadsto \color{blue}{-\frac{c}{b}} \]

   2. distribute-frac-neg2 66.0%

      \[\leadsto \color{blue}{\frac{c}{-b}} \]

   3. expm1-log1p-u 60.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c}{-b}\right)\right)} \]

   4. expm1-undefine 46.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c}{-b}\right)} - 1} \]

9. Applied egg-rr 46.2%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c}{-b}\right)} - 1} \]
10. Step-by-step derivation

    1. sub-neg 46.2%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c}{-b}\right)} + \left(-1\right)} \]

    2. metadata-eval 46.2%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{c}{-b}\right)} + \color{blue}{-1} \]

    3. +-commutative 46.2%

       \[\leadsto \color{blue}{-1 + e^{\mathsf{log1p}\left(\frac{c}{-b}\right)}} \]

    4. log1p-undefine 46.2%

       \[\leadsto -1 + e^{\color{blue}{\log \left(1 + \frac{c}{-b}\right)}} \]

    5. rem-exp-log 51.9%

       \[\leadsto -1 + \color{blue}{\left(1 + \frac{c}{-b}\right)} \]

    6. distribute-frac-neg2 51.9%

       \[\leadsto -1 + \left(1 + \color{blue}{\left(-\frac{c}{b}\right)}\right) \]

    7. unsub-neg 51.9%

       \[\leadsto -1 + \color{blue}{\left(1 - \frac{c}{b}\right)} \]

11. Simplified 51.9%

    \[\leadsto \color{blue}{-1 + \left(1 - \frac{c}{b}\right)} \]

12. Taylor expanded in c around 0 3.2%

    \[\leadsto -1 + \color{blue}{1} \]

13. Final simplification 3.2%

    \[\leadsto 0 \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024143 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))